Optimality of a two-threshold feedback control for a manufacturing system with a production dependent failure rate

We consider a Markovian, failure prone, single machine, single part-type, fluid flow, manufacturing system. The failure rate depends on the production rate through a non decreasing, piecewise constant function which may assume two values. The problem is to minimize a long term average expected cost which penalizes both the presence of waiting customers and the inventory surplus. We derive and prove the optimality of a policy which depends on two thresholds. The expression of these thresholds and of the optimal cost is also included in this note together with some numerical examples.